Secant variety

In algebraic geometry, the secant variety $\mathrm{Sect}(V)$, or the variety of chords, of a projective variety $V\subset {\mathbb{\mathbb{P}}}^r$ is the Zariski closure of the union of all secant lines (chords) to V in ${\mathbb{\mathbb{P}}}^r$:

$\mathrm{Sect}(V)=\bigcup _{x,y\in V}\overline{xy}$

(for $x=y$, the line $\overline{xy}$ is the tangent line.) It is also the image under the projection $p_3:({\mathbb{\mathbb{P}}}^r{)}^3\to {\mathbb{\mathbb{P}}}^r$ of the closure Z of the incidence variety

$\{(x,y,r)|x\wedge y\wedge r=0\}$.

Note that Z has dimension $2\dim V+1$ and so $\mathrm{Sect}(V)$ has dimension at most $2\dim V+1$.

More generally, the $k^{th}$ secant variety is the Zariski closure of the union of the linear spaces spanned by collections of k+1 points on $V$. It may be denoted by ${\mathrm{\Sigma }}_k$. The above secant variety is the first secant variety. Unless ${\mathrm{\Sigma }}_k={\mathbb{\mathbb{P}}}^r$, it is always singular along ${\mathrm{\Sigma }}_{k-1}$, but may have other singular points.

If $V$ has dimension d, the dimension of ${\mathrm{\Sigma }}_k$ is at most $kd+d+k$. A useful tool for computing the dimension of a secant variety is Terracini's lemma.

Examples

A secant variety can be used to show the fact that a smooth projective curve can be embedded into the projective 3-space ${\mathbb{\mathbb{P}}}^3$ as follows. Let $C\subset {\mathbb{\mathbb{P}}}^r$ be a smooth curve. Since the dimension of the secant variety S to C has dimension at most 3, if $r>3$, then there is a point p on ${\mathbb{\mathbb{P}}}^r$ that is not on S and so we have the projection ${\pi }_p$ from p to a hyperplane H, which gives the embedding ${\pi }_p:C\hookrightarrow H\simeq {\mathbb{\mathbb{P}}}^{r-1}$. Now repeat.

If $S\subset {\mathbb{\mathbb{P}}}^5$ is a surface that does not lie in a hyperplane and if $\mathrm{Sect}(S)\ne {\mathbb{\mathbb{P}}}^5$, then S is a Veronese surface.

References

• Eisenbud, David; Joe, Harris (2016), 3264 and All That: A Second Course in Algebraic Geometry, C. U.P., ISBN 978-1107602724 
• Griffiths, P.; Harris, J. (1994). Principles of Algebraic Geometry. Wiley Classics Library. Wiley Interscience. p. 617. ISBN 0-471-05059-8. 
• Joe Harris, Algebraic Geometry, A First Course, (1992) Springer-Verlag, New York. ISBN 0-387-97716-3